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Angle of Friction.                           561

to the surface, but if the surface be rough, the reaction is inclined
to the normal by the friction angle, in a direction away from the
/#//P, and the latter must now be increased by Fn in order to
move the body. If not on the point of sliding, the obliquity of R
may be anything less, down to zero. Two cases are shewn for the
inclined plane, P being directed up or down the plane, but its
value may always be found by force diagram. In moving up the
plane, total pull must balance gravity + Fn, but in moving down
the plane must balance Fn - gravity.

Example 57.— A road engine weighs 12 tons. Find (i) tractive
force of engine to pull 48 tons behind it on a level road, and (2) the
load drawn up a I in 10 incline. Coefficient of traction =150 Ibs.
per ton.

(1)        Tractive force =(12 + 48) 150 = 9000 Ibs.

(2)        P x length = W x height     and P = W x -^
Also R will be found to = -995 W

Tractive force to balance gravity =       —~    = 224     Ibs. per ton.
Tractive force to overcome friction = 150 x '995 = 149*25 ,,         „
Total tractive force =                     373*25 „        „

But the engine only exerts 9000 Ibs.

.*. Total load on incline including engine = -— — = 24*11 tons.
and Load drawn exclusive of engine = 24*11 — 12 = 12*11 tons.

Diminishing Friction by Lubrication.—Spongy metals
like cast iron, brass, and white metal decrease frictional resistance
considerably, but the best results are only obtained by the appli-
cation of unguents.

Lubricants may be solid, as blacklead; semi-solid, as greases
and fats; and liquid, as oils. f Body' for support, and fluidity
to avoid resistance, are both essential requisites, and a careful
choice must be made between extremes. The following are the
unguents used for various purposes:—